Algebraic expressions

Adding \(p+q=6\) and \(p-q=2\) gives \(2p=8\), so \(p=4\). Substituting \(p=4\) into \(p-q=2\) gives \(q=2\). Hence, \(3p-q=3(4)-2=10\). The value \(12\) is only \(3p\); \(q\) must also be subtracted. Exam tip: Add or subtract a pair of linear equations first to eliminate one variable.

To subtract the second polynomial, change the sign of each of its terms: \((5a^2-2ab+3b^2)-(2a^2+5ab-b^2)=5a^2-2ab+3b^2-2a^2-5ab+b^2\). Combining like terms gives \(3a^2-7ab+4b^2\). In option B, the sign of the \(ab\) term is incorrect. Exam tip: when a minus sign comes before brackets, reverse the signs of every term inside the brackets.

The terms of 2x² + 5x + 3 are 2x², 5x, and 3. Here, 3 is the constant term because it contains no x. Removing 3 leaves 2x² + 5x, which has exactly two terms and no constant term. If 5x were removed, the constant term 3 would still remain. Exam tip: A term with no variable is called a constant term.

Substituting the given values, \(x^2y^2=(-1)^2\times2^2=1\times4=4\), while \(2xy=2\times(-1)\times2=-4\). Hence, \(x^2y^2-2xy=4-(-4)=8\). The option 4 results from missing the rule for subtracting a negative number. Exam tip: when a negative term is subtracted, change it to addition.

On expanding, \(2(3x-y)=6x-2y\) and \(4(x+2y)=4x+8y\). Therefore, \(6x-2y+4x+8y-5x=(6+4-5)x+(-2+8)y=5x+6y\). In option B, the sign of the \(y\)-term is incorrect because \(-2y+8y=+6y\). Exam tip: after removing brackets, combine terms with the same variable separately.

The total degree of the monomial \(x^3y^2\) is the sum of the exponents of all its variables: \(3+2=5\). The power of \(y\) is \(2\). Therefore, the required difference is \(5-2=3\). Note that \(5\) is the total degree, not the difference. Exam tip: To find the total degree of a monomial, add the exponents of every variable in it.

Given \(x^2+x=20\). Taking 3 common from the first two terms, \(3x^2+3x-4=3(x^2+x)-4\). Substituting the given value gives \(3(20)-4=60-4=56\). Therefore, 56 is correct. Option 60 is a close distractor because it results from forgetting to subtract 4. Exam tip: Rewrite the expression to contain the exact group given in the question before substituting.

Combining like terms gives \(7x^2y+4x^2y=11x^2y\) and \(-3xy+2xy=-xy\). Therefore, the simplified expression is \(11x^2y-xy-6\), which does not contain the term \(5xy\). The close distractor \(-xy\) does remain because the coefficients of the \(xy\) terms add to \(-3+2=-1\), not \(5\). Exam tip: Add or subtract only those terms that have exactly the same variables raised to the same powers.

Taking the common factor \(xy\) from the first expression gives \(x^2y+xy^2=xy(x+y)\). Therefore, the two expressions are equal for every value of \(x\) and \(y\). Substituting the given values, \(2^2\times1+2\times1^2=6\) and \(2\times1\times(2+1)=6\). Hence, neither expression is greater. Exam tip: factor out the common term \(xy\) to check such equivalence quickly.

Combine like terms: \(8x^2-3x^2=5x^2\), \(-6xy+9xy=3xy\), and \(y^2-5y^2=-4y^2\). Hence, the simplified expression is \(5x^2+3xy-4y^2\). In option B, the coefficients of the \(x^2\) and \(y^2\) terms have been combined incorrectly. Exam tip: add or subtract only terms with the same variables raised to the same powers.

When a whole polynomial is subtracted, the minus sign must be distributed to every term inside its brackets. Starting with \(4x^2-7x+6-(2x^2-5x-3)\), change the signs in the second bracket: \(-(2x^2-5x-3)=-2x^2+5x+3\). The expression becomes \(4x^2-7x+6-2x^2+5x+3\).

Now combine like terms. The quadratic terms give \(4x^2-2x^2=2x^2\), the linear terms give \(-7x+5x=-2x\), and the constants give \(6+3=9\). Hence the simplified expression is \(2x^2-2x+9\), which is option A. Options B and C result from failing to combine or change signs correctly, while option D has the wrong sign for the linear term. The supplied answer and explanation are correct.

Substitution gives (16+2(-8)-3(4)+5=-7). Even and odd powers of a negative number affect signs differently.

Only like terms can be added or subtracted. Here, 9a^2b and -6a^2b are like terms, so they give 3a^2b. Similarly, -4ab^2 and 11ab^2 give 7ab^2. Hence, the simplified expression is 3a^2b+7ab^2. Option A results from adding the coefficients of a^2b incorrectly. Exam tip: Before combining terms, check that both the variables and their exponents are identical.

displaystyle (2k+1)x-4x=[(2k+1)-4]x=(2k-3)x. Since its simplified form is 9x, the coefficients of x must be equal: 2k-3=9. Thus, 2k=12 and k=6. For example, choosing 5 gives coefficient 7, not 9. Exam tip: while combining like terms, add or subtract their coefficients only.

The equations are \(x+y=11\) and \(x-y=5\). Subtract the second equation from the first: \((x+y)-(x-y)=11-5\), so \(2y=6\). Hence, \(y=3\). Option 5 is the constant in \(x-y=5\), not the value of \(y\). Exam tip: While adding or subtracting simultaneous equations, carefully track the signs to eliminate the required variable.

Expand the brackets: 5(2p-3q)=10p-15q and -3(p+4q)=-3p-12q. Thus, the expression becomes 10p-15q-3p-12q+2q. Combining like terms gives (10p-3p)+(−15q−12q+2q)=7p−25q. Therefore, option A is correct. Remember that the minus sign before 3(p+4q) affects both terms inside the bracket.

Given \(m=-1\) and \(n=4\), we have \(m^2=(-1)^2=1\) and \(n^2=4^2=16\). Thus, \(3m^2n=3\times1\times4=12\), while \(-2mn^2=-2\times(-1)\times16=32\). Therefore, the value of the expression is \(12+32=44\). Option 32 is only the value of the second term, not of the complete expression. Exam tip: squaring a negative number gives a positive result.

The governing concept is correct translation of operation order into algebra. “The difference of x and y” is written as (x − y), and “the square of the difference” requires the whole quantity to be squared: (x − y)^2. Adding 4xy to it gives (x − y)^2 + 4xy, so option D is correct. Option A expands only part of the idea and incorrectly uses x^2 − y^2, which is not the square of a difference. Option B squares y alone, while option C subtracts the squared difference from 4xy, reversing the stated order.

Only like terms can be added or subtracted. The terms containing x^2y are 7x^2y and -9x^2y, so their sum is (7-9)x^2y=-2x^2y. Therefore, the coefficient of x^2y is -2. The terms 3xy^2 and 5xy^2 are not like x^2y; they add to 8xy^2 instead. Exam tip: like terms must have the same variables raised to the same powers.

First simplify the innermost bracket: \(3x-(5x-8)=3x-5x+8=-2x+8\). Then \(6x-(-2x+8)=6x+2x-8=8x-8\). Therefore, \(8x-8\) is the correct option. In \(8x+8\), the sign of the final constant is incorrect because the outer minus changes the sign of every term inside the bracket. Exam tip: When a bracket is preceded by a minus sign, change the signs of all terms inside it.